Skip to main content
#
Full text of "Treatise On Analysis Vol-Ii"

252 XIV INTEGRATION IN LOCALLY COMPACT GROUPS * + «> f(st)dt rxi^i==rcci^isdt= Jo t Jo st which proves our assertion. (1423) Let Gbea locally compact group with neutral element e, and let iibea (left orriqht) Haar measure on G. Then G is discrete if and only if n({e}) > 0; and G is compact if and only (/>*(G) < + oo (i.e., if and only if » is a bounded measure (13.20)). It is clear that if G is compact then p is bounded. If G is discrete, {e} is an open neighborhood of e, and hence /*({*}) > 0 because the support of A* is the whole of G Conversely, let V be a compact neighborhood of e. If /*({*}) > 0, we have u({s}) = i*({e}) for all seG, because \L is invariant; hence the number of points of V is finite and ^OO/MW)- Since G is Hausdorff it follows that G is discrete. Suppose now that /* is bounded and (say) left-invariant. Consider the set (E of finite subsets [sl9 s2, ... ,sn] in G such that st V n Sj V = 0 whenever i^j. We have n;£(V) = ju(^V u j2 V u - - • u sn V) ^ /i(G), hence 72^XG)/MOO- Hence G contains a subset {^,...,^n} having the largest possible number of elements. For each j e G it then follows that *V must meet at least one of the st V, that is to say, s e st VV J. Hence G is the union of the sets ^VV""1, which are compact (12.10.4), and so G is compact. (14.2.4) Let G be a locally compact group, V an open subset of G, and ju a nonzero positive measure on V, having the following property: ifUis an open subset ofV and if seG is such that sU c V, then the image of the measure liv, induced by fJionU (13.1.8), under the homeomorphism x\-+sx (13.1.6), is the measure /*sU induced by n on s\J. Then there exists a unique left Haar measure a on G which induces \i on V. For each s e G, let p, be the image of \i under the homeomorphism xl_> sx of V onto jV. The restriction of jus to V n ^V is the image of /v iv n v under the restriction of xt-*sx to s'^V n V. By hypothesis, this image is /*vnsv- BY translation it follows that for all s, tin G the measures jus and ^ have the same restriction to ^V n rV. By virtue of (13.1.9), there exists a positive measure a on G which induces /*, on sV for all s e G. Clearly a is left- invariant, and is therefore the unique left Haar measure on G which induces /i onV.< b, containing the support of /. Hence for each interval [c, d] in